In the following, let G be a finitegroup that acts on a setX. For each g in G let Xg denote the set of elements in X that are fixed byg. Burnside's lemma asserts the following formula for the number of orbits, denoted |X/G|:[2]

Thus the number of orbits (a natural number or +∞) is equal to the average number of points fixed by an element of G (which is also a natural number or infinity). If G is infinite, the division by |G| may not be well-defined; in this case the following statement in cardinal arithmetic holds:

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The number of rotationally distinct colourings of the faces of a cube using three colours can be determined from this formula as follows.

Let X be the set of 36 possible face colour combinations that can be applied to a cube in one particular orientation, and let the rotation group G of the cube act on X in the natural manner. Then two elements of X belong to the same orbit precisely when one is simply a rotation of the other. The number of rotationally distinct colourings is thus the same as the number of orbits and can be found by counting the sizes of the fixed sets for the 24 elements of G.

Cube with coloured faces

one identity element which leaves all 36 elements of X unchanged

six 90-degree face rotations, each of which leaves 33 of the elements of X unchanged

three 180-degree face rotations, each of which leaves 34 of the elements of X unchanged

eight 120-degree vertex rotations, each of which leaves 32 of the elements of X unchanged

six 180-degree edge rotations, each of which leaves 33 of the elements of X unchanged

William Burnside stated and proved this lemma, attributing it to Frobenius 1887 in his 1897 book on finite groups. But, even prior to Frobenius, the formula was known to Cauchy in 1845. In fact, the lemma was apparently so well known that Burnside simply omitted to attribute it to Cauchy. Consequently, this lemma is sometimes referred to as the lemma that is not Burnside's.[3] (see also Stigler's law of eponymy). This is less ambiguous than it may seem: Burnside contributed many lemmas to this field.